This context of this question is Rutten's Universal Coalgebra, used for modelling systems. I'm interested in finding a description of a functor between different types of coalgebras corresponding to finding a certain subcoalgebra.

An $F+1$-coalgebra $\langle S,\alpha:S\to F(S)+1\rangle$ can be thought of as a system whose transition shape is given by functor F plus the possibility of an error/termination, given by the +1. Assume that $F$ is a so-called Kripke polynomial functor: $F::=Id ~|~ B ~|~ F+F ~|~ F\times F ~|~ F^ A ~|~ \mathcal{P}_\omega F$, thus it preserves pullbacks.

A subcoalgebra of $\langle S,\alpha:S\to F(S)+1\rangle$ is coalgebra $\langle S',\alpha':S'\to F(S')+1\rangle$, where $S'\subseteq S$ and $\alpha'$ is $\alpha'$ restricted to $S'$, such that its range falls within $F(S')+1$.

I want to find the maximal subcoalgebra of this coalgebra which corresponds to an $F$-coalgebra. In terms of my application, this means I'm looking for the subset of states $S'\subseteq S$ that do not lead to the error state.

Clearly, I can take the pullback of the functions $\alpha:S\to F(S)+1$ and $inl:F(S)\to F(S)+1$ to get a set $S_0\subseteq S$ which do not lead to an error in the first step. Iterating this process for functors $F^i+1$ seems to lead to progressively smaller subsets $S_i$ of $S$ each avoiding the error state for $i$ steps. What I'm lacking is a coherent description of the process.

Is whether there is a more universal description of this construction in terms of limits or colimits, or at least, some known approaches to the problem?

I asked this question on the computer science theory overflow (here), but am re-asking it here as I received no feedback.

finitary, i.e. preserve filtered colimits, rather than preserving pullbacks? $\newcommand{\Pw}{\mathcal{P}_\omega} \Pw$ doesn't preserve pullbacks, if I'm not mistaken: the map $\Pw(2 \times_1 2) \rightarrow \Pw(2) \times_{\Pw(1)} \Pw(2)$ isn't injective, since $\{(x,y),(x',y')\}$ and $\{(x,y'),(x',y)\}$ both get sent to $(\{x,x'\},\{y,y'\})$. – Peter LeFanu Lumsdaine Sep 8 '10 at 14:42